Group homomorphism/Kernel/Definition

Let ${\displaystyle {}G}$ and ${\displaystyle {}H}$ be groups, and let

${\displaystyle \varphi \colon G\longrightarrow H}$

be a group homomorphism. Then the preimage of the neutral element is called the kernel of ${\displaystyle {}\varphi }$, denoted by

${\displaystyle {}\operatorname {kern} \varphi =\varphi ^{-1}(e_{H})={\left\{g\in G\mid \varphi (g)=e_{H}\right\}}\,.}$